Modeling a Periodic Signal Using Fourier Series

This paper covers the concept of Fourier series and its application for a periodic signal. A periodic signal is a signal that repeats its pattern over time at regular intervals. The idea inspiring is to approximate a regular periodic signal, under Dirichlet conditions, via a linear superposition of trigonometric functions, thus Fourier polynomials are constructed. The Dirichlet conditions, are a set of mathematical conditions, providing a foundational framework for the validity of the Fourier series representation. By understanding and applying these conditions, we can accurately represent and process periodic signals, leading to advancements in various areas of signal processing. The resulting Fourier approximation allows complex periodic signals to be expressed as a sum of simpler sinusoidal functions, making it easier to analyze and manipulate such signals.

考虑衬砌截面协调变形约束的既有隧道受盾构下穿施工影响的Fourier能量变分解

目前关于新建盾构开挖诱发既有地铁隧道变形的理论研究,大多将既有隧道简化为不含接头的Euler-Bernoulli梁,而忽视了衬砌截面的剪切变形与管片接头的局部刚度削弱。此外,既有地基模型理论较少考虑衬砌管片与接触土体的协调变形约束。首先,采用Loganathan-Polous公式计算盾构开挖诱发的土体自由位移场,并将既有隧道视为考虑剪切变形的Timoshenko梁,基于沿轴线变化的抗弯与剪切刚度函数来描述管片接头的局部削弱作用;然后,结合能量变分原理与连续弹性体Mindlin解,建立考虑衬砌截面协同变形约束的隧道位移控制方程,将盾构开挖诱发的附加荷载施加于既有隧道,并使用Fourier级数方法求解既有隧道纵向变形;最后,将Fourier能量变分解与3组工程实测数据进行对比验证,取得了较好的一致性。此外,考虑隧道抗弯刚度、剪切刚度、接头刚度削弱作用、土体参数与隧道空间位置等因素的共同影响,使用抗弯系数、剪切系数及接头系数进行敏感性分析。结果表明,考虑截面变形协调约束的理论解更为符合实际,不考虑变形协调约束所得隧道位移结果偏大;衬砌接头局部刚度的削弱作用将造成隧道弯矩沿轴线在接头处发生明显锯齿状突变;抗弯系数与剪切系数的增长均导致隧道结构所受弯矩降低;剪切系数由低到高的增长将导致隧道变形模式发生“弯曲控制-弯剪混合-剪切控制”三阶段变化;隧道变形模式进入“剪切控制”阶段的过程中,抗弯刚度的变化对隧道内力影响逐渐降低;接头系数的增长将造成隧道所受弯矩降低,但隧道抗弯刚度较大时其影响逐渐减弱。

ASYMPTOTIC BEHAVIOR OF ECKHOFF'S METHOD FOR FOURIER SERIES CONVERGENCE ACCELERATION

The current paper considers the problem of recovering a function using a limited number of its Fourier coefficients. Specifically, a method based on Bernoulli-like polynomials suggested and developed by Krylov, Lanczos, Gottlieb and Eckhoff is examined. Asymptotic behavior of approximate calculation of the so-called ＂jumps＂ is studied and asymptotic L2 constants of the rate of convergence of the method are computed.

Convergence Properties of Generalized Fourier Series on a Parallel Hexagon Domain

A new Rogosinski-type kernel function is constructed using kernel function of partial sums Sn（f; t） of generalized Fourier series on a parallel hexagon domain Ω associating with threedirection partition. We prove that an operator Wn（f; t） with the new kernel function converges uniformly to any continuous function f（t） ∈ Cn（Ω） （the space of all continuous functions with period Ω） on Ω. Moreover, the convergence order of the operator is presented for the smooth approached function.

Alternative Fourier Series Expansions with Accelerated Convergence

The key objective of this paper is to improve the approximation of a sufficiently smooth nonperiodic function defined on a compact interval by proposing alternative forms of Fourier series expansions. Unlike in classical Fourier series, the expansion coefficients herein are explicitly dependent not only on the function itself, but also on its derivatives at the ends of the interval. Each of these series expansions can be made to converge faster at a desired polynomial rate. These results have useful implications to Fourier or harmonic analysis, solutions to differential equations and boundary value problems, data compression, and so on.

CONVERGENCE OF LOGARITHMIC MEANS OF MULTIPLE WALSH-FOURIER SERIES

Norlund logarithmic means of multiple Walsh-Fourier series acting from space L In^d-1 L （[0, 1）d）, d≥1 into space weak - LI（[0,1）^d） are studied. The maximal Orlicz space such that the Norlund logarithmic means of multiple Walsh-Fourier series for the functions from this space converge in d-dimensional measure is found.

APPROXIMATION OF CONVEX TYPE FUNCTION BY PARTIAL SUMS OF FOURIER SERIES

The concept of convex type function is introduced in this paper,from which a kin d of convex decomposition approach is proposed.As one of applications of this a pproach,the approximation of the convex type function by the partial sum of its Fourier series is inves tigated.Moreover,the order of approximation is describe d with the 2th continuous modulus.

Estimation of permafrost thermal behavior using Fourier series model

Permafrost,being an important component of the cryosphere,is sensitive to climate change.Therefore,it is necessary to investigate the change of temperature within permafrost.In this study,we proposed a Fourier series model derived from the conduction equation to simulate permafrost thermal behavior over a year.The boundary condition was represented by the Fourier series and the geothermal gradient.The initial condition was represented as a linear function relative to the geothermal gradient.A comparative study of the different models(sinusoidal model,Fourier series model,and the proposed model)was conducted.Data collected from the northern Da Xing’anling Mountains,Northeast China,were applied for parameterization and validation for these models.These models were compared with daily mean ground temperature from the shallow permafrost layer and annual mean ground temperature from the bottom permafrost layer,respectively.Model performance was assessed using three coefficients of accuracy,i.e.,the mean bias error,the root mean square error,and the coefficient of determination.The comparison results showed that the proposed model was accurate enough to simulate temperature variation in both the shallow and bottom permafrost layer as compared with the other two Fourier series models(sinusoidal model and Fourier model).The proposed model expanded on a previous Fourier series model for which the initial and bottom boundary conditions were restricted to being constant.

Estimation of Winds at Different Isobaric Levels Based on the Observed Winds at 850 hPa Level Using Double Fourier Series

A technique based on the double Fourier series is developed to estimate the winds at different isobaric levels forthe limited area domain, 35°E to 140°E and 30°S to 40°N, using the observed winds at 850 hPa lcvcl for the month ofJune. For this purpose the wind field at a level under consideration is taken in the ratio form with that of 850 hPa level and the coefficients of the double Fouricr series are computed. These coefficients are subsequently used to computethe winds which are compared with the actual winds. The results of the double Fourier series technique are comparedwith those of the polynomial surface fitting method developed by Bavadekar and Khaladkar (1 992). The technique isalso applied for the daily wind data of 11. June, 1979 and the validation of the technique is tested for a few radiosondestations of india. The computed winds for these radiosonde stations arc quite close to observed winds.

APPROXIMATION OF SPHERICAL MEANS OF MULTIPLE FOURIER INTEGRALS AND FOURIER SERIES ON SET OF FULL MEASURE

In this paper we consider the approximation for functions in some subspaces of L^2 by spherical means of their Fourier integrals and Fourier series on set of full measure. Two main theorems are obtained.

Approximation by Nrlund Means of Hexagonal Fourier Series

Let f be an H-periodic HOlder continuous function of two real variables.The error ｜｜f-Nn （p;f）｜｜ is estimated in the uniform norm and in the Holder norm,where p=（pk）k=0∞is a nonincreasing sequence of positive numbers and Nn （p;f） is thenth Norlund mean of hexagonal Fourier series of f with respect to p = （pk）k∞=0.

CONVERGENCE OF DOUBLE WALSH-FOURIER SERIES AND HARDY SPACES

It is proved that the maximal operator of the Marczinkiewicz-Fejér meams of a double Walsh-Fourier series is bounded from the two-dimensional dyadic martingale Hardy space H p to L p (2/3<p<∞) and is of weak type (1,1). As a consequence we obtain that the Marczinkiewicz-Fejér means of a function f∈L 1 converge a.e. to the function in question. Moreover, we prove that these means are uniformly bounded on H p whenever 2/3<p<∞. Thus, in case f∈H p , the Marczinkiewicz-Fejér means conv f in H p norm. The same results are proved for the conjugate means, too.

UNIFORM CONVERGENCE OF CES■RO MEANS OF NEGATIVE ORDER OF DOUBLE TRIGONOMETRIC FOURIER SERIES

In this paper we prove that if f ∈ C （[-π, π]^2） and the function f is bounded partial p-variation for some p ∈[1, ＋∞）, then the double trigonometric Fourier series of a function f is uniformly （C;-α,-β） summable （α＋β 〈 1/p,α,β 〉 0） in the sense of Pringsheim. If α ＋ β ≥ 1/p, then there exists a continuous function f0 of bounded partial p-variation on [-π,π]^2 such that the Cesàro （C;-α,-β） means σn,m^-α,-β（f0;0,0） of the double trigonometric Fourier series of f0 diverge over cubes.

Rainfall Forecasting Using Fourier Series

The need for accurate rainfall prediction is readily apparent when considering many benefits in which such information would provide for river control, reservoir operation, forestry interests, flood mitigation, etc.. Due to importance of rainfall in many aspects, studies on rainfall forecast have been conducted since a few decades ago. Although many methods have been introduced, all the researches describe the study as complex because it involves numerous variables and still need to be improved. Nowadays, there are various traditional techniques and mathematical models available, yet, there are no result on which method provide the most reliable estimation. AR （auto-regressive）, ARMA （auto-regressive moving average）, ARIMA （auto-regressive integrated moving average） and ANNs （artificial neural networks） were introduced as a useful and efficient tool for modeling and forecasting. The conventional time series provide reasonable accuracy but suffer from the assumptions of stationary and linearity. The concept of neurons was introduced first which then developed to ANNs with back propagation training algorithm. Although certain ANNs） models are equivalent to time series model, but it is limited to short term forecasting. This Paper presents a mathematical approach for rainfall forecasting for Iran on monthly basic. The model is trained for monthly rainfall forecasting and tested to evaluate the performance of the model. The result Shows reasonably good accuracy for monthly rainfall forecasting.

SUMMATION OF TRIGONOMETRIC SERIES BY FOURIER TRANSFORMS

This paper gives the theorems concerning the summation of trigonometric series with the help of Fourier transforms. By means of the known results of Fourier transforms, many difficult and complex problems of summation of trigonometric series can be solved. This method is a comparatively unusual way to find the summation of trigonometric series, and has been used to establish the comprehensive table of summation of trigonometric series. In this table 10 thousand scries arc given, and most of them are new.

On the Efficacy of Fourier Series Approximations for Pricing European Options

This paper investigates several competing procedures for computing the prices of vanilla European options, such as puts, calls and binaries, in which the underlying model has a characteristic function that is known in semi-closed form. The algorithms investigated here are the half-range Fourier cosine series, the half-range Fourier sine series and the full-range Fourier series. Their performance is assessed in simulation experiments in which an analytical solution is available and also for a simple affine model of stochastic volatility in which there is no closed-form solution. The results suggest that the half-range sine series approximation is the least effective of the three proposed algorithms. It is rather more difficult to distinguish between the performance of the half-range cosine series and the full-range Fourier series. However there are two clear differences. First, when the interval over which the density is approximated is relatively large, the full-range Fourier series is at least as good as the half-range Fourier cosine series, and outperforms the latter in pricing out-of-the-money call options, in particular with maturities of three months or less. Second, the computational time required by the half-range Fourier cosine series is uniformly longer than that required by the full-range Fourier series for an interval of fixed length. Taken together, these two conclusions make a case for pricing options using a full-range range Fourier series as opposed to a half-range Fourier cosine series if a large number of options are to be priced in as short a time as possible.

Further Discussion on the Calculation of Fourier Series

Fourier series is an important mathematical concept. It is well known that we need too much computation to expand the function into Fourier series. The existing literature only pointed that its Fourier series is sine series when the function is an odd function and its Fourier series is cosine series when the function is an even function. And on this basis, in this paper, according to the function which satisfies different conditions, we give the different forms of Fourier series and the specific calculation formula of Fourier coefficients, so as to avoid unnecessary calculation. In addition, if a function is defined on [0,a], we can make it have some kind of nature by using the extension method as needed. So we can get the corresponding form of Fourier series.

Fourier-Series Representation of Discontinuous Functions and Its Physical Applications

In this work, Fourier-series representation of a discontinuous function is used to highlight and clarify the controversial problem of finding the value of the function at a point of discontinuity. Several physical situations are presented to examine the consequences of this kind of representation and its impact on some widely well-known problems whose results are not clearly understood or justified.

On the Equiconvergence of the Fourier Series and Integral of Distributions

We prove equiconvergence of the Bochner-Riesz means of the Fourier series and integral of distributions with compact support from the Liouville spaces.

Application of Fourier Series Expansion Method with PMLs to the Microcavities on Two-Dimensional Photonic Crystals

By using a Fourier series expansion method combined with Chew＇s perfectly matched layers （PMLs）, we analyze the frequency and quality factor of a micro-cavity on a two-dimensional photonic crystal is analyzed. Compared with the results by the method without PML and finite-difference time-domain （FDTD） based on supercell approximation, it can be shown that by the present method with PMLs, the resonant frequency and the quality factor values can be calculated satisfyingly and the characteristics of the micro-cavity can be obtained by changing the size and permittivity of the point defect in the micro-cavity.